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A Block Reduction Method for Random Band Matrices with General Variance Profiles

Jiaqi Fan, Fan Yang, Jun Yin

TL;DR

The paper develops a block-reduction framework for random band matrices with general variance profiles, establishing bulk delocalization in 1D and 2D under near-optimal band-width conditions and extending to Wegner orbital models with precise interaction strength scaling. Central to the approach is a block-structured flow of variance profiles that restores a self-consistent loop hierarchy, together with a random-walk representation for Theta-propagators and deterministic bounds on K-loops. By combining global resolvent laws with a detailed loop-hierarchy analysis and a flow-based propagation to local scales, the authors prove sharp local laws, isotropic delocalization results, and optimal bounds on localization lengths in the generalized setting. The block reduction methodology thus provides a flexible and robust tool to handle broad variance profiles and inter-block couplings, bridging previous BlockRBM results and Wegner-model analyses while clarifying the limits of current techniques in higher dimensions.

Abstract

We present a novel block reduction method for the study of a general class of random band matrices (RBM) defined on the $d$-dimensional lattice $\mathbb{Z}_{L}^d:=\{1,2,\ldots,L\}^{d}$ for $d\in \{1,2\}$, with band width $W$ and an almost arbitrary variance profile subject to a core condition. We prove the delocalization of bulk eigenvectors for such RBMs under the assumptions $W\ge L^{1/2+\varepsilon}$ in one dimension and $W\geq L^{\varepsilon}$ in two dimensions, where $\varepsilon$ is an arbitrarily small constant. This result extends the findings of arXiv:2501.01718 and arXiv:2503.07606 on block RBMs to models with general variance profiles. Furthermore, we generalize our results to Wegner orbital models with small interaction strength $λ\ll 1$. Under the sharp condition $λ\gg W^{-d/2}$, we establish optimal lower bounds for the localization lengths of bulk eigenvectors, thereby extending the results of arXiv:2503.11382 to settings with nearly arbitrary potential and hopping terms. Our block reduction method provides a powerful and flexible framework that reduces both the dynamical analysis of the loop hierarchy and the derivation of deterministic estimates for general RBMs to the corresponding analysis of block RBMs, as developed in arXiv:2501.01718, arXiv:2503.07606 and arXiv:2503.11382.

A Block Reduction Method for Random Band Matrices with General Variance Profiles

TL;DR

The paper develops a block-reduction framework for random band matrices with general variance profiles, establishing bulk delocalization in 1D and 2D under near-optimal band-width conditions and extending to Wegner orbital models with precise interaction strength scaling. Central to the approach is a block-structured flow of variance profiles that restores a self-consistent loop hierarchy, together with a random-walk representation for Theta-propagators and deterministic bounds on K-loops. By combining global resolvent laws with a detailed loop-hierarchy analysis and a flow-based propagation to local scales, the authors prove sharp local laws, isotropic delocalization results, and optimal bounds on localization lengths in the generalized setting. The block reduction methodology thus provides a flexible and robust tool to handle broad variance profiles and inter-block couplings, bridging previous BlockRBM results and Wegner-model analyses while clarifying the limits of current techniques in higher dimensions.

Abstract

We present a novel block reduction method for the study of a general class of random band matrices (RBM) defined on the -dimensional lattice for , with band width and an almost arbitrary variance profile subject to a core condition. We prove the delocalization of bulk eigenvectors for such RBMs under the assumptions in one dimension and in two dimensions, where is an arbitrarily small constant. This result extends the findings of arXiv:2501.01718 and arXiv:2503.07606 on block RBMs to models with general variance profiles. Furthermore, we generalize our results to Wegner orbital models with small interaction strength . Under the sharp condition , we establish optimal lower bounds for the localization lengths of bulk eigenvectors, thereby extending the results of arXiv:2503.11382 to settings with nearly arbitrary potential and hopping terms. Our block reduction method provides a powerful and flexible framework that reduces both the dynamical analysis of the loop hierarchy and the derivation of deterministic estimates for general RBMs to the corresponding analysis of block RBMs, as developed in arXiv:2501.01718, arXiv:2503.07606 and arXiv:2503.11382.

Paper Structure

This paper contains 20 sections, 24 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Main results
  4. Key ideas
  5. Preliminaries
  6. Flow
  7. Loop hierarchy and primitive loops
  8. Propagators and evolution kernels
  9. Deterministic analysis
  10. Properties of primitive loops
  11. Proof of \ref{['lemma_properties_of_propagators']}
  12. Global laws
  13. Proof of \ref{['lemma_global_law']}
  14. Proof of \ref{['improved_global_laws']}
  15. Resolvent entry estimates
  16. ...and 5 more sections

Key Result

Theorem 1.5

Consider the matrix $H$ defined in def_considered_model with $d\in\{1,2\}$. Let $\kappa,\delta>0$ be arbitrary small constants. If $W\ge L^{\delta}$, then for any constants $\tau,D>0$, the following estimate holds for sufficiently large $L$: where $\eta_*$ is defined as

Theorems & Definitions (52)

  • Definition 1.1: Block structure
  • Definition 1.2: Random band matrix
  • Example 1.3: Translationally invariant random band matrices
  • Example 1.4: Wegner orbital model with general block variance profiles
  • Theorem 1.5: Delocalization
  • Theorem 1.6: Local law
  • Theorem 1.7: Quantum unique ergodicity
  • Theorem 1.8: Quantum diffusion
  • Definition 1.9: Stochastic domination and high probability event
  • Lemma 2.1
  • ...and 42 more